volume | PIER Journals

Abstract

The magnetic field of an axially symmetric coil or magnetic material system can be computed by expansion of the central and remote zonal harmonics, using the Legendre polynomials. This method can be 100-1000 times faster than the more widely known elliptic integral method and is more general than the similar radial series expansion. We present the zonal harmonic method for field, scalar and vector potential calculation of circular current loops, of general axisymmetric coils and magnetic materials, and of special coils with rectangular cross section, with various source representations: currents, magnetic dipoles and equivalent magnetic charges. We discuss in detail the convergence properties of the zonal harmonic expansions, and we show the generalization of the method for special three-dimensional magnetic systems.

1. Paszkowski, B., Electron Optics, London Eliffe Books, 1968.

2. Szilágyi, M., Electron and Ion Optics, Plenum Press, 1988.

3. Hawkes, P. W. and E. Kasper, Principles of Electron Optics, Vol. 1, Academic Press, Harcourt Brace Jovanovich, 1989.

4. Montgomery, D. B. and R. J. Weggel, Solenoid Magnet Design, Robert E. Krieger Publ. Comp., 1980.

5. Saint-Jalmes, H., J. Taquin, and Y. Barjhoux, "Optimization of homogeneous electromagnetic coil systems: Application to whole-body NMR imaging magnets," Rev. Sci. Instrum., Vol. 52, 1501, 1981.
doi:10.1063/1.1136484

6. Williams, J. E. C., "Superconducting magnets for MRI," IEEE Trans. Nucl. Sci., Vol. 31, 994, 1984.
doi:10.1109/TNS.1984.4333424

7. Sanger, P. A., "Present status of MRI magnets at Oxford," IEEE Trans. Magnetics, Vol. 21, 436, 1985.
doi:10.1109/TMAG.1985.1063856

8. Morad, R., et al. "Way of improving the stability and homogeneity of MRI magnets," IEEE Trans. Magnetics, Vol. 24, 1282, 1988.
doi:10.1109/20.11471

9. Vetter, J., G. Ries, and T. Reichert, "A 4-tesla superconducting whole-body magnet for MR imaging and spectroscopy," IEEE Trans. Magnetics, Vol. 24, 1285, 1988.
doi:10.1109/20.11472

10. Shimada, Y., et al. "Superconducting magnet with self-shield for whole body magnetic resonance imaging," IEEE Trans. Magnetics, Vol. 27, 1685, 1991.
doi:10.1109/20.133512

11. Kraus, C., et al. "Final results from phase II of the Mainz neutrino mass search in tritium β decay," Eur. Phys. J. C, Vol. 40, 447, 2005.
doi:10.1140/epjc/s2005-02139-7

12. Lobashev, V. M., "The search for the neutrino mass by direct method in the tritium beta-decay and perspectives of study it in the project KATRIN," Nucl. Phys. A, Vol. 719, 153c, 2003.
doi:10.1016/S0375-9474(03)00985-0

13. Glück, F., et al. "The neutron decay retardation spectrometer aSPECT: Electromagnetic design and systematic effects," Eur. Phys. J. A, Vol. 23, 135, 2005.
doi:10.1140/epja/i2004-10057-1

14. Baessler, S., et al. "First measurements with the neutron decay spectrometer aSPECT," Eur. Phys. J. A, Vol. 38, 17, 2008.
doi:10.1140/epja/i2008-10660-0

15. Beck, M., et al. "WITCH: A recoil spectrometer for weak interaction and nuclear physics studies," Nucl. Instrum. Methods A, 503-567, 2003.

16. Friedag, P., Bahnverfolgungssimulationen für das WITCH experiment, Diploma thesis, University of Münster, 2008.

17. Angrik, J., et al. KATRIN design report 2004, FZKA Scientific Report 7090, Forschungszentrum Karlsruhe, 2005, http://bibliothek.fzk.de/zb/berichte/FZKA7090.pdf.

18. Maxwell, J. C., A Treatise on Electricity and Magnetism, Vol. 1, Clarendon Press, 1873.

19. Durand, E., Magnetostatique, Masson et Cie, 1968.

20. Jackson, J. D., Classical Electrodynamics, John Wiley & Sons, 1999.

21. Smythe, W. R., Static and Dynamic Electricity, McGraw Hill Book Company, 1968.

22. Kellogg, O. D., Foundations of Potential Theory, Springer Verlag, 1967.

23. Garrett, M. W., "Axially symmetric systems for generating and measuring magnetic fields," J. Appl. Phys., Vol. 22, 1091, 1951.
doi:10.1063/1.1700115

24. Garrett, M. W., "The method of zonal harmonics," High Magnetic Fields, H. Kolm, et al. (eds.), John Wiley and Sons, 1962.

25. Garrett, M. W., Computer programs using zonal harmonics for magnetic properties of current systems with special reference to the IBM 7090, ORNL-3318, Oak Ridge National Laboratory, USA, 1962.

26. Garrett, M. W., "Thick cylindrical coil systems for strong magnetic fields with field or gradient homogeneities of the 6th to 20th order," J. Appl. Phys., Vol. 38, 2563, 1967.
doi:10.1063/1.1709950

27. Garrett, M. W., "Calculation of fields, forces, and mutual inductances of current systems by elliptic integrals," J. Appl. Phys., Vol. 34, 2567, 1963.
doi:10.1063/1.1729771

28. Franzen, W., "Generation of uniform magnetic fields by means of air-core coils," Rev. Sci. Instrum., Vol. 33, 933, 1962.
doi:10.1063/1.1718031

29. Marshall, H. L. and H. E. Weaver, "Application of the Garrett method to calculation of coil geometries for generating homogeneous magnetic fields in superconducting solenoids," J. Appl. Phys., Vol. 34, 3175, 1963.
doi:10.1063/1.1729158

30. Girard, B. and M. Sauzade, "Calcul des solenoides compenses du 6eme ordre a volume de bobinage minimum," Nucl. Instrum. Methods, Vol. 25, 269, 1964.
doi:10.1016/0029-554X(63)90198-8

31. Kaminishi, K. and S. Nawata, "Practical method of improving the uniformity of magnetic fields generated by single and double Helmholtz coils," Rev. Sci. Instrum., Vol. 52, 447, 1981.
doi:10.1063/1.1136609

32. Glück, F., "Axisymmetric electric field calculation with zonal harmonic expansion," Progress In Electromagnetics Research B, Vol. 32, 319-350, 2011.
doi:10.2528/PIERB11042106

33. Flatt, B., Designstudien für das KATRIN Experiment, Diploma thesis, University of Mainz, 2001.

34. Thümmler, T., Entwicklung von Methoden zur Untergrun-dreduzierung am Mainzer Tritium-β-Spektrometer, Diploma thesis, University of Mainz, 2002.

35. Müller, B., Umbau des Mainzer Neutrinomassenexperiments und Untergrunduntersuchungen im Hinblick auf KATRIN, Diploma thesis, University of Mainz, 2002.

36. Essig, K., Untersuchungen zur Penningfalle zwischen den Spektrometern des KATRIN Experiments, Diploma thesis, University of Bonn, 2004.

37. Thümmler, T., Präzisionsüberwachung und Kalibration der Hochspannung für das KATRIN Experiment, Dissertation, University of Münster, 2007.

38. Pocanic, D., et al. "Nab: Measurement principles, apparatus and uncertainties," Nucl. Instrum. Methods A, Vol. 611, 211, 2009.
doi:10.1016/j.nima.2009.07.065

39. Dubbers, D., et al. "A clean, bright, and versatile source of neutron decay products," Nucl. Instrum. Methods A, Vol. 596, 238, 2008.
doi:10.1016/j.nima.2008.07.157

40. Valerius, K., Elektromagnetisches Design für das Hauptspektrometer des KATRIN Experiments, Diploma thesis, University of Bonn, 2004.

41. Valerius, K., Spectrometer-related background processes and their suppression in the KATRIN experiment, Dissertation, University of MÄunster, 2009.

42. Hugenberg, K., Design of the electrode system for the KATRIN main spectrometer, Diploma thesis, University of Münster, 2008.

43. Zacher, M., Electromagnetic design and field emission studies for the inner electrode system of the KATRIN main spectrometer, Diploma thesis, University of Münster, 2009.

44. Wandkowsky, N., Design and background simulations for the KATRIN main spectrometer and air coil system, Diploma thesis, Karlsruhe Institute of Technology, 2009.

45. Fränkle, F., Background investigations of the KATRIN Pre-spectrometer, Dissertation, Karlsruhe Institute of Technology, 2010.

46. Sturm, M., Bestimmung der Tritiumflussreduktion einer Tritium-Argon-Frostpumpe für das Neutrinomassenexperiment KATRIN, Diploma thesis, Karlsruhe Institute of Technology, 2007.

47. Groh, S., Untersuchung von UV-Laser induziertem Untergrund am KATRIN Vorspektrometer, Diploma thesis, Karlsruhe Institute of Technology, 2010.

48. Reimer, S., Ein elektrostatisches Dipolsystem zur Eliminierung von Ionen in der DPS2-F des KATRIN Experimentes, Diploma thesis, Karlsruhe Institute of Technology, 2009.

49. Hötzel, M., Berechnung von KATRIN Messspektren unter Einbeziehung der Eigenschaften der fensterlosen gasförmigen Tritiumquelle, Diploma thesis, Karlsruhe Institute of Technology, 2009.

50. Schwarz, J., Design zur Messung der elektro-optischen Eigen-schaften der differentiellen Pumpstrecke DPS2-F des KATRIN Experiments, Diploma thesis, Karlsruhe Institute of Technology, 2010.

51. Valerius, K., et al. "Prototype of an angular-selective photoelectron calibration source for the KATRIN-experiment," J. Inst., Vol. 6, P01002, 2011.
doi:10.1088/1748-0221/6/01/P01002

52. Lukic, S., et al. "Ion source for tests of ion behavior in the KATRIN beam line," Rev. Sci. Instrum., Vol. 82, 013303, 2011.
doi:10.1063/1.3504372

53. KATRIN homepage, Talks and Publications, Diploma and Ph.D. theses, http://www-ik.fzk.de/~katrin/publications/thesis.html.

54. Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, AG Prof. Dr. C. Weinheimer, http://www.unimuenster.de/Physik.KP/AGWeinheimer/Arbeiten-de.html.

55. Vöcking, S. Implementierung der multipole boundary element methode für das KATRIN-experiment, Diploma thesis, University of Münster, 2008.

56. Corona, T. J. Tools for electromagnetic field simulation in the KATRIN experiment, Master thesis, MIT, 2009.

57. Leiber, B. Non-axially symmetric field and trajectory calculations for the KATRIN experiment, Diploma thesis, Karlsruhe Institute of Technology, 2010.

58. Babutzka, M., et al. The Comprehensive Guide to KASSIOPEIA, Version 1.00.00, KATRIN internal report.

59. Cowan, E. W., Basic Electromagnetism, Academic Press, 1968.

60. Panofsky, W. K. H. and M. Phillips, Classical Electricity and Magnetism, Dover Publications, 1962.

61. Wangsness, R. K., Electromagnetic Fields, John Wiley & Sons, 1979.

62. Ravaud, R., G. Lemarquand, V. Lemarquand, and C. Depollier, "Discussion about the analytical calculation of the magnetic field created by permanent magnets," Progress In Electromagnetics Research B, Vol. 11, 281-297, 2009.
doi:10.2528/PIERB08112102

63. Ravaud, R. and G. Lemarqand, "Comparison of the Coulombian and Amperian current models for calculating the magnetic field produced by radially magnetized arc-shaped permanent magnets," Progress In Electromagnetics Research, Vol. 95, 309, 2009.
doi:10.2528/PIER09042105

64. Zisserman, A., R. Saunders, and J. Caldwell, "Analytic solutions for axisymmetric magnetostatic systems involving iron," IEEE Trans. Magnetics, Vol. 23, 3895, 1987.
doi:10.1109/TMAG.1987.1065773

65. Saunders, R., A. Zisserman, and C. J. McCauley, "The calculation of magnetostatic fields from axisymmetric conductors," J. Phys. D, Vol. 29, 533, 1996.
doi:10.1088/0022-3727/29/3/008

66. Ravaud, R., et al. "Mutual inductance and force exerted between thick coils," Progress In Electromagnetics Research, Vol. 102, 367, 2010.
doi:10.2528/PIER10012806

67. Gardner, M. E., et al. "Production of a uniform magnetic field by means of an end-corrected solenoid," Rev. Sci. Instrum., Vol. 31, 929, 1960.
doi:10.1063/1.1717109

68. Andrew, E. R., I. Roberts, and R. C. Gupta, "Helmholtz-type coils of finite cross section," J. Sci. Instrum., Vol. 43, 936, 1966.
doi:10.1088/0950-7671/43/12/315

Dr. Ferenc Gluck